Modern Analytical Cheymistry - Chapter 7 pdf

Figure 7.1Percent of overall variance s2 due to the method as a function of the relative magnitudes of the standard deviation of the method and the standard deviation of sampling sm/ss.

179

Obtaining and Preparing

Samples for Analysis

W hen we first use an analytical method to solve a problem, it is

not unusual to find that our results are of questionable accuracy or so

imprecise as to be meaningless Looking back we may find that

nothing in the method seems amiss In designing the method we

considered sources of determinate and indeterminate error and took

appropriate steps, such as including a reagent blank and calibrating

our instruments, to minimize their effect Why, then, might a carefully

designed method give such poor results? One explanation is that we

may not have accounted for errors associated with the sample When

we collect the wrong sample or lose analyte while preparing the sample

for analysis, we introduce a determinate source of error If we do not

collect enough samples or collect samples of the wrong size, the

precision of the analysis may suffer In this chapter we consider how

collecting samples and preparing them for analysis can affect the

accuracy and precision of our results.

Figure 7.1

Percent of overall variance (s2 ) due to the

method as a function of the relative

magnitudes of the standard deviation of the

method and the standard deviation of

sampling (sm/ss ) The dotted lines show that

the variance due to the method accounts for

10% of the overall variance when

ss= 3 ×sm.

*Values for t can be found in Appendix 1B.

7A The Importance of Sampling

When a manufacturer produces a chemical they wish to list as ACS Reagent Grade,they must demonstrate that it conforms to specifications established by the Ameri-can Chemical Society (ACS) For example, ACS specifications for NaHCO3requirethat the concentration of iron be less than or equal to 0.001% w/w To verify that aproduction lot meets this standard, the manufacturer performs a quantitative analy-sis, reporting the result on the product’s label Because it is impractical to analyzethe entire production lot, its properties are estimated from a limited sampling Sev-eral samples are collected and analyzed, and the resulting mean,X, and standard de-–

viation, s, are used to establish a confidence interval for the production lot’s true

mean, µ

7.1

where n is the number of samples, and t is a statistical factor whose value is

deter-mined by the number of samples and the desired confidence level.*

Selecting a sample introduces a source of determinate error that cannot be rected during the analysis If a sample does not accurately represent the populationfrom which it is drawn, then an analysis that is otherwise carefully conducted willyield inaccurate results Sampling errors are introduced whenever we extrapolatefrom a sample to its target population To minimize sampling errors we must col-lect the right sample

cor-Even when collecting the right sample, indeterminate or random errors in pling may limit the usefulness of our results Equation 7.1 shows that the width of a

sam-confidence interval is directly proportional to the standard deviation

The overall standard deviation for an analysis, so, is determined by

ran-dom errors affecting each step of the analysis For convenience, we vide the analysis into two steps Random errors introduced when collect-

di-ing samples are characterized by a standard deviation for sampldi-ing, ss The standard deviation for the analytical method, sm, accounts for ran-

dom errors introduced when executing the method’s procedure The

re-lationship among so, ss, and smis given by a propagation of random error

s2o = s2m + s2s 7.2

Equation 7.2 shows that an analysis’ overall variance may be ited by either the analytical method or sample collection Unfortu-nately, analysts often attempt to minimize overall variance by im-proving only the method’s precision This is futile, however, if the standarddeviation for sampling is more than three times greater than that for themethod.1 Figure 7.1 shows how the ratio sm/ssaffects the percentage of overallvariance attributable to the method When the method’s standard deviation isone third of that for sampling, indeterminate method errors explain only 10%

lim-of the overall variance Attempting to improve the analysis by decreasing sm

provides only a nominal change in the overall variance

EXAMPLE 7.1

A quantitative analysis for an analyte gives a mean concentration of 12.6 ppm

The standard deviation for the method is found to be 1.1 ppm, and that due to

sampling is 2.1 ppm (a) What is the overall variance for the analysis? (b) By

how much does the overall variance change if smis improved by 10% to 0.99

ppm? (c) By how much does the overall variance change if ssis improved by

10% to 1.9 ppm?

SOLUTION

(a) The overall variance is

so= sm2+ ss2= (1.1)2+ (2.1)2= 1.21 + 4.41 = 5.62 ≈ 5.6(b) Improving the method’s standard deviation changes the overall variance to

so= (0.99)2+ (2.1)2= 0.98 + 4.41 = 5.39 ≈ 5.4Thus, a 10% improvement in the method’s standard deviation changes theoverall variance by approximately 4%

(c) Changing the standard deviation for sampling

so= (1.1)2+ (1.9)2= 1.21 + 3.61 = 4.82 ≈ 4.8

improves the overall variance by almost 15% As expected, since ssis larger

than sm, a more significant improvement in the overall variance is realizedwhen we focus our attention on sampling problems

To determine which step has the greatest effect on the overall variance, both sm2

and s2smust be known The analysis of replicate samples can be used to estimate the

overall variance The variance due to the method is determined by analyzing a

stan-dard sample, for which we may assume a negligible sampling variance The variance

due to sampling is then determined by difference

EXAMPLE 7.2

The following data were collected as part of a study to determine the effect of

sampling variance on the analysis of drug animal-feed formulations.2

The data on the left were obtained under conditions in which random errors in

sampling and the analytical method contribute to the overall variance The data

on the right were obtained in circumstances in which the sampling variance is

known to be insignificant Determine the overall variance and the

contributions from sampling and the analytical method

sampling plan

A plan that ensures that a representative

sample is collected.

SOLUTION

The overall variance, so, is determined using the data on the left and is equal to

4.71 × 10–7 The method’s contribution to the overall variance, sm2, isdetermined using the data on the right and is equal to 7.00×10–8 The variance

due to sampling, ss2, is therefore

ss2= so – sm2= 4.71×10–7– 7.00×10–8= 4.01×10–7

7B Designing A Sampling Plan

A sampling plan must support the goals of an analysis In characterization studies a

sample’s purity is often the most important parameter For example, a material entist interested in the surface chemistry of a metal is more likely to select a freshlyexposed surface, created by fracturing the sample under vacuum, than a surface thathas been exposed to the atmosphere for an extended time In a qualitative analysisthe sample’s composition does not need to be identical to that of the substancebeing analyzed, provided that enough sample is taken to ensure that all componentscan be detected In fact, when the goal of an analysis is to identify components present at trace levels, it may be desirable to discriminate against major componentswhen sampling In a quantitative analysis, however, the sample’s composition mustaccurately represent the target population The focus of this section, therefore, is ondesigning a sampling plan for a quantitative analysis

sci-Five questions should be considered when designing a sampling plan:

1 From where within the target population should samples be collected?

2 What type of samples should be collected?

3 What is the minimum amount of sample needed for each analysis?

4 How many samples should be analyzed?

5 How can the overall variance be minimized?

Each of these questions is considered below in more detail

7B.1 Where to Sample the Target Population

Sampling errors occur when a sample’s composition is not identical to that of thepopulation from which it is drawn When the material being sampled is homoge-neous, individual samples can be taken without regard to possible sampling errors.Unfortunately, in most situations the target population is heterogeneous in eithertime or space As a result of settling, for example, medications available as oral sus-pensions may have a higher concentration of their active ingredients at the bottom

of the container Before removing a dose (sample), the suspension is shaken to imize the effect of this spatial heterogeneity Clinical samples, such as blood orurine, frequently show a temporal heterogeneity A patient’s blood glucose level, forinstance, will change in response to eating, medication, or exercise Other systemsshow both spatial and temporal heterogeneities The concentration of dissolved O2

min-in a lake shows a temporal heterogeneity due to the change min-in seasons, whereaspoint sources of pollution may produce a spatial heterogeneity

When the target population’s heterogeneity is of concern, samples must be quired in a manner that ensures that determinate sampling errors are insignificant

ac-If the target population can be thoroughly homogenized, then samples can be takenwithout introducing sampling errors In most cases, however, homogenizing the

random sample

A sample collected at random from the target population.

target population is impracticable Even more important, homogenization destroys

information about the analyte’s spatial or temporal distribution within the target

population

Random Sampling The ideal sampling plan provides an unbiased estimate of the

target population’s properties This requirement is satisfied if the sample is collected

at random from the target population.3Despite its apparent simplicity, a true

ran-dom sample is difficult to obtain Haphazard sampling, in which samples are

col-lected without a sampling plan, is not random and may reflect an analyst’s

uninten-tional biases The best method for ensuring the collection of a random sample is to

divide the target population into equal units, assign a unique number to each unit,

and use a random number table (Appendix 1E) to select the units from which to

sample Example 7.3 shows how this is accomplished

EXAMPLE 7.3

To analyze the properties of a 100 cm×100 cm polymer sheet, ten 1 cm×1 cm

samples are to be selected at random and removed for analysis Explain how a

random number table can be used to ensure that samples are drawn at random

SOLUTION

As shown in the following grid, we divide the polymer sheet into 10,000 1 cm×

1 cm squares, each of which can be identified by its row number and its

column number

For example, the highlighted square is in row 1 and column 2 To pick ten

squares at random, we enter the random number table at an arbitrary point,

and let that number represent the row for the first sample We then move

through the table in a predetermined fashion, selecting random numbers for

the column of the first sample, the row of the second sample, and so on until all

ten samples have been selected Since our random number table (Appendix 1E)

uses five-digit numbers we will use only the last two digits Let’s begin with the

fifth entry and use every other entry after that The fifth entry is 65423 making

the first row number 23 The next entry we use is 41812, giving the first column

number as 12 Continuing in this manner, the ten samples are as follows:

98 99

A randomly collected sample makes no assumptions about the target tion, making it the least biased approach to sampling On the other hand, randomsampling requires more time and expense than other sampling methods since agreater number of samples are needed to characterize the target population.

popula-Judgmental Sampling The opposite of random sampling is selective, or

judg-mental sampling, in which we use available information about the target

popula-tion to help select samples Because assumppopula-tions about the target populapopula-tion areincluded in the sampling plan, judgmental sampling is more biased than randomsampling; however, fewer samples are required Judgmental sampling is commonwhen we wish to limit the number of independent variables influencing the re-sults of an analysis For example, a researcher studying the bioaccumulation ofpolychlorinated biphenyls (PCBs) in fish may choose to exclude fish that are toosmall or that appear diseased Judgmental sampling is also encountered in manyprotocols in which the sample to be collected is specifically defined by the regula-tory agency

Systematic Sampling Random sampling and judgmental sampling represent tremes in bias and the number of samples needed to accurately characterize the tar-

ex-get population Systematic sampling falls in between these extremes In systematic

sampling the target population is sampled at regular intervals in space or time For asystem exhibiting a spatial heterogeneity, such as the distribution of dissolved O2in

a lake, samples can be systematically collected by dividing the system into discreteunits using a two- or three-dimensional grid pattern (Figure 7.2) Samplesare collected from the center of each unit, or at the intersection of gridlines When a heterogeneity is time-dependent, as is common in clinicalstudies, samples are drawn at regular intervals

When a target population’s spatial or temporal heterogeneity shows aperiodic trend, a systematic sampling leads to a significant bias if samplesare not collected frequently enough This is a common problem whensampling electronic signals, in which case the problem is known as alias-ing Consider, for example, a signal consisting of a simple sine wave Fig-ure 7.3a shows how an insufficient sampling frequency underestimates thesignal’s true frequency

According to the Nyquist theorem, to determine a periodic signal’s true

fre-quency, we must sample the signal at a rate that is at least twice its frequency ure 7.3b); that is, the signal must be sampled at least twice during a single cycle orperiod When samples are collected at an interval of ∆t, the highest frequency thatcan be accurately monitored has a frequency of (2∆t)–1 For example, if samples arecollected every hour, the highest frequency that we can monitor is 0.5 h–1, or a peri-odic cycle lasting 2 h A signal with a cycling period of less than 2 h (a frequency ofmore than 0.5 h–1) cannot be monitored Ideally, the sampling frequency should be

(Fig-at least three to four times th(Fig-at of the highest frequency signal of interest Thus, if

an hourly periodic cycle is of interest, samples should be collected at least every15–20 min

Systematic–Judgmental Sampling Combinations of the three primary approaches

to sampling are also possible.4One such combination is systematic–judgmental

sampling, which is encountered in environmental studies when a spatial or

tempo-judgmental sampling

Samples collected from the target

population using available information

about the analyte’s distribution within

the population.

Nyquist theorem

Statement that a periodic signal must be

sampled at least twice each period to

avoid a determinate error in measuring

its frequency.

systematic–judgmental sampling

A sampling plan that combines

judgmental sampling with systematic

sampling.

Figure 7.2

Example of a systematic sampling plan for

collecting samples from a lake Each solid

dot represents a sample collected from

within the sampling grid.

systematic sampling

Samples collected from the target

population at regular intervals in time or

space.

stratified sampling

A sampling plan that divides the population into distinct strata from which random samples are collected.

ral distribution of pollutants is anticipated For example, a plume of waste

leaching from a landfill can reasonably be expected to move in the same

di-rection as the flow of groundwater The systematic–judgmental sampling

plan shown in Figure 7.4 includes a rectangular grid for systematic sampling

and linear transects extending the sampling along the plume’s suspected

major and minor axes.5

Stratified Sampling Another combination of the three primary

approaches to sampling is judgmental–random, or stratified sampling.

Many target populations are conveniently subdivided into distinct units,

or strata For example, in determining the concentration of particulate

Pb in urban air, the target population can be subdivided by particle size

In this case samples can be collected in two ways In a random sampling,

differences in the strata are ignored, and individual samples are collected

at random from the entire target population In a stratified sampling

the target population is divided into strata, and random samples are

collected from within each stratum Strata are analyzed separately, and

their respective means are pooled to give an overall mean for the target

population

The advantage of stratified sampling is that the composition of each

stra-tum is often more homogeneous than that of the entire target population

When true, the sampling variance for each stratum is less than that when the

target population is treated as a single unit As a result, the overall sampling

variance for stratified sampling is always at least as good as, and often better

than, that obtained by simple random sampling

Convenience Sampling One additional method of sampling deserves

brief mention In convenience sampling, sample sites are selected using

criteria other than minimizing sampling error and sampling variance In

a survey of groundwater quality, for example, samples can be collected

by drilling wells at randomly selected sites, or by making use of existing

wells The latter method is usually the preferred choice In this case, cost,

expedience, and accessibility are the primary factors used in selecting

sam-pling sites

7B.2 What Type of Sample to Collect

After determining where to collect samples, the next step in designing a sampling

plan is to decide what type of sample to collect Three methods are commonly used

to obtain samples: grab sampling, composite sampling, and in situ sampling The

most common type of sample is a grab sample, in which a portion of the target

population is removed at a given time and location in space A grab sample,

there-fore, provides a “snapshot” of the target population Grab sampling is easily

adapted to any of the sampling schemes discussed in the previous section If the

tar-get population is fairly uniform in time and space, a set of grab samples collected at

random can be used to establish its properties A systematic sampling using grab

samples can be used to characterize a target population whose composition varies

over time or space

A composite sample consists of a set of grab samples that are combined to

form a single sample After thoroughly mixing, the composite sample is analyzed.Because information is lost when individual samples are combined, it is normallydesirable to analyze each grab sample separately In some situations, however, thereare advantages to working with composite samples One such situation is in deter-mining a target population’s average composition over time or space For example,wastewater treatment plants are required to monitor and report the average compo-sition of treated water released to the environment One approach is to analyze a se-ries of individual grab samples, collected using a systematic sampling plan, and av-erage the results Alternatively, the individual grab samples can be combined toform a single composite sample Analyzing a single composite sample instead ofmany individual grab samples, provides an appreciable savings in time and cost.Composite sampling is also useful when a single sample cannot supply sufficientmaterial for an analysis For example, methods for determining PCBs in fish oftenrequire as much as 50 g of tissue, an amount that may be difficult to obtain from asingle fish Tissue samples from several fish can be combined and homogenized,and a 50-g portion of the composite sample taken for analysis

A significant disadvantage of grab samples and composite samples is the need toremove a portion of the target population for analysis As a result, neither type of sam-ple can be used to continuously monitor a time-dependent change in the target popu-

lation In situ sampling, in which an analytical sensor is placed directly in the target

population, allows continuous monitoring without removing individual grab samples.For example, the pH of a solution moving through an industrial production line can becontinually monitored by immersing a pH electrode within the solution’s flow

Figure 7.4

Systematic–judgmental sampling scheme for

monitoring the leaching of pollutants from a

landfill Sites where samples are collected

are represented by the solid dots.

Direction of groundwater flow

Sampling done within the population

without physically removing the sample.

*See Chapter 4 to review the properties of a binomial distribution.

7B.3 How Much Sample to Collect

To minimize sampling errors, a randomly collected grab sample must be of an

ap-propriate size If the sample is too small its composition may differ substantially

from that of the target population, resulting in a significant sampling error Samples

that are too large, however, may require more time and money to collect and

ana-lyze, without providing a significant improvement in sampling error

As a starting point, let’s assume that our target population consists of two types

of particles Particles of type A contain analyte at a fixed concentration, and type B

particles contain no analyte If the two types of particles are randomly distributed,

then a sample drawn from the population will follow the binomial distribution.* If

we collect a sample containing n particles, the expected number of particles

con-taining analyte, nA, is

nA= np where p is the probability of selecting a particle of type A The sampling standard

deviation is

7.3

The relative standard deviation for sampling, ss,r, is obtained by dividing equation

7.3 by nA

Solving for n allows us to calculate the number of particles that must be sampled to

obtain a desired sampling variance

7.4

Note that the relative sampling variance is inversely proportional to the number of

particles sampled Increasing the number of particles in a sample, therefore,

im-proves the sampling variance

EXAMPLE 7.4

Suppose you are to analyze a solid where the particles containing analyte

represent only 1×10–7% of the population How many particles must be

collected to give a relative sampling variance of 1%?

SOLUTION

Since the particles of interest account for 1×10–7% of all particles in the

population, the probability of selecting one of these particles is only 1×10–9

Substituting into equation 7.4 gives

Thus, to obtain the desired sampling variance we need to collect 1×1013

A sample containing 1013particles can be fairly large Suppose this is equivalent

to a mass of 80 g Working with a sample this large is not practical; but does thismean we must work with a smaller sample and accept a larger relative samplingvariance? Fortunately the answer is no An important feature of equation 7.4 is thatthe relative sampling variance is a function of the number of particles but not theircombined mass We can reduce the needed mass by crushing and grinding the par-ticles to make them smaller Our sample must still contain 1013particles, but sinceeach particle is smaller their combined mass also is smaller If we assume that a par-ticle is spherical, then its mass is proportional to the cube of its radius

Mass ∝r3Decreasing a particle’s radius by a factor of 2, for example, decreases its mass by a fac-tor of 23, or 8 Instead of an 80-g sample, a 10-g sample will now contain 1013particles

EXAMPLE 7.5

Assume that the sample of 1013particles from Example 7.4 weighs 80 g By howmuch must you reduce the radius of the particles if you wish to work with asample of 0.6 g?

SOLUTION

To reduce the sample from 80 g to 0.6 g you must change its mass by a factor of

This can be accomplished by decreasing the radius of the particles by a factor of

Few populations, however, meet the conditions for a true binomial tion Real populations normally contain more than two types of particles, with theanalyte present at several levels of concentration Nevertheless, many well-mixedpopulations, in which the population’s composition is homogeneous on the scale atwhich we sample, approximate binomial sampling statistics Under these conditionsthe following relationship between the mass of a randomly collected grab sample,

distribu-m, and the percent relative standard deviation for sampling, R, is often valid.6

ated by determining R using several samples of similar mass Once Ksis known, the

mass of sample needed to achieve a desired relative standard deviation for sampling

can be calculated

EXAMPLE 7.6

The following data were obtained in a preliminary determination of the

amount of inorganic ash in a breakfast cereal

Determine Ksand the amount of sample needed to give a relative standard

deviation for sampling of ±2.0% Predict the percent relative standard

deviation and the absolute standard deviation if samples of 5 g are collected

SOLUTION

To determine Kswe need to know the average mass of the cereal samples and

the relative standard deviation for the %(w/w) ash The average mass of the five

cereal samples is 1.0007 g The average %(w/w) ash and the absolute standard

deviation are, respectively, 1.298% and 0.03194 The percent relative standard

deviation, therefore, is

Thus

Ks= mR2= (1.0007 g)(2.46)2= 6.06 gThe amount of sample needed to give a relative standard deviation of ±2%,

therefore, is

If we use 5.00-g samples, then the expected percent relative standard deviation is

and the expected absolute standard deviation is

s X

When the target population is segregated, or stratified, equation 7.5 provides a poorestimate of the amount of sample needed to achieve a desired relative standard de-viation for sampling A more appropriate relationship, which can be applied to bothsegregated and nonsegregated samples, has been proposed.7

7.6

where nsis the number of samples to be analyzed, m is the mass of each sample, A is

a homogeneity constant accounting for the random distribution of analyte in the

target population, and B is a segregation constant accounting for the nonrandom

distribution of analyte in the target population Equation 7.6 shows that samplingvariance due to the random distribution of analyte can be minimized by increasingeither the mass of each sample or the total number of samples Sampling errors due

to the nonrandom distribution of the analyte, however, can only be minimized byincreasing the total number of samples Values for the homogeneity constant andheterogeneity constant are determined using two sets of samples that differ signifi-cantly in mass

EXAMPLE 7.7

To develop a sampling plan for the determination of PCBs in lake sediments,the following two experiments are conducted First, 15 samples, each with amass of 1.00 g, are analyzed, giving a sampling variance of 0.0183 In a secondexperiment, ten samples, each with a mass of 10.0 g, are analyzed, giving asampling variance of 0.0069 If samples weighing 5.00 g are to be collected, howmany are needed to give a sampling variance of 0.0100? If five samples are to becollected, how much should each sample weigh?

SOLUTION

Substituting known values for the two experiments into equation 7.6 gives thefollowing pair of simultaneous equations

Solving for A and B gives values of 0.228 and 0.0462, respectively The number

of 5.00-g samples is determined by solving

for n, giving n = 9.2≈ 9 samples When using five samples, the mass of each isgiven by the equation

s

s s

2 = +

7B.4 How Many Samples to Collect

In the previous section we considered the amount of sample needed to minimize

the sampling variance Another important consideration is the number of samples

required to achieve a desired maximum sampling error If samples drawn from the

target population are normally distributed, then the following equation describes

the confidence interval for the sampling error

where ns is the number of samples and ssis the sampling standard deviation

Rear-ranging and substituting e for the quantity (µ–X), gives the number of samples as–

7.7

where ss2and e2are both expressed as absolute uncertainties or as relative

uncertain-ties Finding a solution to equation 7.7 is complicated by the fact that the value of t

depends on ns As shown in Example 7.8, equation 7.7 is solved iteratively.

EXAMPLE 7.8

In Example 7.6 we found that an analysis for the inorganic ash content of a

breakfast cereal required a sample of 1.5 g to establish a relative standard

deviation for sampling of ±2.0% How many samples are needed to obtain a

relative sampling error of no more than 0.80% at the 95% confidence level?

SOLUTION

Because the value of t depends on ns, and the value of nsis not yet known, we

begin by letting ns=∞and use the associated value of t From Appendix 1B,

the value for t is 1.96 Substituting known values into equation 7.7

Letting ns = 24, the value of t from Appendix 1B is 2.075 Recalculating nsgives

When ns = 27, the value of t is 2.066 Recalculating ns, we find

Since two successive calculations give the same value for ns, an iterative

solution has been found Thus, 27 samples are needed to achieve the desired

sampling error

Equation 7.7 only provides an estimate for the smallest number of samples

ex-pected to produce the desired sampling error The actual sampling error may be

substantially higher if the standard deviation for the samples that are collected is

significantly greater than the standard deviation due to sampling used to calculate ns

n t s e

This is not an uncommon problem For a target population with a relative samplingvariance of 50 and a desired relative sampling error of ±5%, equation 7.7 predictsthat ten samples are sufficient In a simulation in which 1000 samples of size 10were collected, however, only 57% of the samples resulted in sampling errors of lessthan ±5%.8By increasing the number of samples to 17 it was possible to ensure thatthe desired sampling error was achieved 95% of the time.

7B.5 Minimizing the Overall Variance

A final consideration in developing a sampling plan is to minimize the overall ance for the analysis Equation 7.2 shows that the overall variance is a function ofthe variance due to the method and the variance due to sampling As we have seen,

vari-we can improve the variance due to sampling by collecting more samples of propersize Increasing the number of times we analyze each sample improves the variance

due to the method If ss2is significantly greater than sm2, then the method’s variancecan be ignored and equation 7.7 used to estimate the number of samples to analyze.Analyzing any sample more than once will not improve the overall variance, sincethe variance due to the method is insignificant

If sm2is significantly greater than ss2, then we only need to collect and analyze a

single sample The number of replicate analyses, nr, needed to minimize the error

due to the method is given by an equation similar to equation 7.7

Unfortunately, the simple situations just described are often the exception Inmany cases, both the sampling variance and method variance are significant, andboth multiple samples and replicate analyses of each sample are required The over-all error in this circumstance is given by

7.8

Equation 7.8 does not have a unique solution because different combinations of nsand nrgive the same overall error The choice of how many samples to collect andhow many times each sample should be analyzed is determined by other concerns,such as the cost of collecting and analyzing samples, and the amount of availablesample

EXAMPLE 7.9

A certain analytical method has a relative sampling variance of 0.40% and arelative method variance of 0.070% Evaluate the relative error (α= 0.05) if(a) you collect five samples, analyzing each twice; and, (b) you collect twosamples, analyzing each five times

SOLUTION

Both sampling strategies require a total of ten determinations Using Appendix 1B,

we find that the value of t is 2.26 Substituting into equation 7.8, we find that the

relative error for the first sampling strategy is

s r

2 2 1 2/

n t s e

r = 2 22m

and that for the second sampling strategy is

As expected, since the relative method variance is better than the relative

sampling variance, a sampling strategy that favors the collection of more

samples and few replicate analyses gives the better relative error

7C Implementing the Sampling Plan

After a sampling plan has been developed, it is put into action Implementing a

sampling plan normally involves three steps: physically removing the sample from

its target population, preserving the sample, and preparing the sample for analysis

Except for in situ sampling, the analysis of a sample occurs after removing it from

the target population Since sampling exposes the target population to potential

contamination, the sampling device must be inert and clean

Once a sample is withdrawn from a target population, there is a danger that it

may undergo a chemical or physical change This is a serious problem since the

properties of the sample will no longer be representative of the target population

For this reason, samples are often preserved before transporting them to the

labora-tory for analysis Even when samples are analyzed in the field, preservation may still

be necessary

The initial sample is called the primary, or gross sample and may be a single

increment drawn from the target population, or a composite of several increments

In many cases the gross sample cannot be analyzed without further treatment

Pro-cessing the gross sample may be used to reduce the sample’s particle size, to transfer

the sample into a more readily analyzable form, or to improve its homogeneity

In the sections that follow, these three steps are considered for the sampling of

liquids (including solutions), gases, and solids

7C.1 Solutions

Typical examples of liquid samples include those drawn from containers of

com-mercial solvents; beverages, such as milk or fruit juice; natural waters, including

from lakes, streams, seawater, and rain; bodily fluids, such as blood and urine; and,

suspensions, such as those found in many oral medications

Sample Collection Homogeneous solutions are easily sampled by siphoning,

de-canting, or by using a pipet or syringe Unfortunately, few solutions are truly

homo-geneous When the material to be sampled is of manageable size, manual shaking is

often sufficient to ensure homogeneity Samples may then be collected with a pipet,

a syringe, or a bottle The majority of solutions, however, cannot be sampled in this

manner To minimize the effect of heterogeneity, the method for collecting the

gross sample must be adapted to the material being sampled

The environmental sampling of waters and wastewaters provides a good

illus-tration of many of the methods used to sample solutions The chemical

composi-tion of surface waters, such as streams, rivers, lakes, estuaries, and oceans, is

influ-enced by flow rate and depth Rapidly flowing shallow streams and rivers, and

shallow (<5 m) lakes are usually well mixed and show little stratification with

depth Grab samples are conveniently collected by submerging a capped bottlebelow the surface and removing the cap The air–water interface, which may be en-riched with heavy metals9or contaminated with oil, is avoided when collecting thesample After the sample bottle is filled, the cap is replaced and the bottle removed.Slowly moving streams and rivers, lakes deeper than 5 m, estuaries, and oceans mayshow substantial stratification Grab samples from near the surface can be collected

as described earlier, whereas samples at greater depths are collected with a weightedsample bottle that is lowered to the desired depth Once it has reached the desireddepth, the sample bottle is opened, allowed to fill, and closed before retrieving.Grab samples can be analyzed individually, giving information about changes in theanalyte’s concentration with depth Alternatively, the grab samples may be pooled

to form a composite sample

Wells used for collecting groundwater samples must be purged before the ple is collected, since the chemical composition of water in the well-casing and inthe adjacent matrix may be significantly different from that of the surroundinggroundwater These differences may result from contaminants introduced whendrilling the well, or differences in the groundwater’s redox potential when exposed

sam-to atmospheric oxygen In general, wells are purged by pumping out a volume ofwater equivalent to several well-casing volumes, or until the water’s temperature,

pH, or specific conductance are constant Samples collected from municipal watersupplies must also be purged since the chemical composition of water left standing

in pipes may differ significantly from the treated water supply Samples are collected

at faucets after flushing the pipes for 2–3 min

Samples from municipal wastewater treatment plants and samples of industrialdischarges often are collected as 24-h composites Samples are obtained using anautomatic sampler that periodically removes individual grab samples The volume

of each sample increment and the frequency of sampling may be constant or mayvary in response to changes in flow rate

Sample containers for collecting solutions are made from glass or plastic tainers made from Kimax or Pyrex brand borosilicate glass have the advantage ofbeing sterilizable, easy to clean, and inert to all solutions except those that arestrongly alkaline The disadvantages of glass containers are cost, weight, and thelikelihood of breakage Plastic containers are made from a variety of polymers, in-cluding polyethylene, polypropylene, polycarbonate, polyvinyl chloride, and Teflon(polytetrafluoroethylene) Plastic containers are lightweight, durable, and, exceptfor those manufactured from Teflon, inexpensive In most cases glass or plastic bot-tles may be used, although polyethylene bottles are generally preferred because oftheir lower cost Glass containers are always used when collecting samples for theanalysis of pesticides, oil and grease, and organics because these species often inter-act with plastic surfaces Since glass surfaces easily adsorb metal ions, plastic bottlesare preferred when collecting samples for the analysis of trace metals

Con-In most cases the sample bottle has a wide mouth, making it easy to fill and move the sample A narrow-mouth sample bottle is used when exposing the sample

re-to the container cap or re-to the outside environment is undesirable Unless exposure

to plastic is a problem, caps for sample bottles are manufactured from polyethylene.When polyethylene must be avoided, the container cap includes an inert interiorliner of neoprene or Teflon

Sample Preservation Once removed from its target population, a liquid sample’schemical composition may change as a result of chemical, biological, or physicalprocesses Following its collection, samples are preserved by controlling the solu-

tion’s pH and temperature, limiting its exposure to light or to the atmosphere, or by

adding a chemical preservative After preserving, samples may be safely stored for

later analysis The maximum holding time between preservation and analysis

de-pends on the analyte’s stability and the effectiveness of sample preservation Table 7.1

provides a list of sample preservation methods and maximum holding times for

sev-eral analytes of importance in the analysis of water and wastewater

Sample Preparation Most analytical methods can be applied to analytes in a liquid

or solution state For this reason a gross sample of a liquid or solution does not

need additional processing to bring it into a more suitable form for analysis

Typical examples of gaseous samples include automobile exhaust, emissions from

industrial smokestacks, atmospheric gases, and compressed gases Also included

with gaseous samples are solid aerosol particulates

Sample Collection The simplest approach for collecting a gas sample is to fill a

container, such as a stainless steel canister or a Tedlar/Teflon bag, with a portion of

the gas A pump is used to pull the gas into the container, and, after flushing the

container for a predetermined time, the container is sealed This method has the

ad-vantage of collecting a more representative sample of the gas than other collection

techniques Disadvantages include the tendency for some gases to adsorb to the

container’s walls, the presence of analytes at concentrations too low to detect with

accuracy and precision, and the presence of reactive gases, such as ozone and

nitro-gen oxides, that may change the sample’s chemical composition with time, or react

with the container When using a stainless steel canister many of these

disadvan-tages can be overcome with cryogenic cooling, which changes the sample from a

gaseous to a liquid state

Due to the difficulty of storing gases, most gas samples are collected using

ei-ther a trap containing a solid sorbent or by filtering Solid sorbents are used to

col-lect volatile gases (vapor pressures more than approximately 10–6 atm) and

semi-volatile gases (vapor pressures between approximately 10–6atm and 10–12atm), and

filtration is used to collect nonvolatile gases

Table 7.1 Preservation Methods and Maximum Holding Times

for Selected Water and Wastewater Parameters

organochlorine pesticides 1 mL 10 mg/mL HgCl2; 7 days without extraction

or addition of extracting 40 days with extraction solvent

Solid sorbent sampling is accomplished by passing the gas through a canisterpacked with sorbent particles Typically 2–100 L of gas is sampled when collectingvolatile compounds, and 2–500 m3when collecting semivolatile gases.* A variety ofinorganic, organic polymer and carbon sorbents have been used Inorganic sor-bents, such as silica gel, alumina, magnesium aluminum silicate, and molecularsieves, are efficient collectors for polar compounds Their efficiency for collectingwater, however, limits their sorption capacity for many organic compounds.

Organic polymeric sorbents are manufactured using polymeric resins of

2,4-diphenyl-p-phenylene oxide or styrene-divinylbenzene for volatile compounds,

or polyurethane foam for semivolatile compounds These materials have a low ity for water and are efficient collectors for all but the most highly volatile organiccompounds and some low-molecular-weight alcohols and ketones The adsorbingability of carbon sorbents is superior to that of organic polymer resins Thus, carbonsorbents can be used to collect those highly volatile organic compounds that cannot

affin-be collected by polymeric resins The adsorbing ability of carbon soraffin-bents may affin-be adisadvantage, however, since the adsorbed compounds may be difficult to desorb.Nonvolatile compounds are normally present either as solid particulates orbound to solid particulates Samples are collected by pulling large volumes of gasthrough a filtering unit where the particulates are collected on glass fiber filters.One of the most significant problems with sorbent sampling is the limited ca-pacity of the sorbent to retain gaseous analytes If a sorbent’s capacity is exceededbefore sampling is complete then a portion of the analyte will pass through the can-ister without being retained, making an accurate determination of its concentrationimpossible For this reason it is not uncommon to place a second sorbent canisterdownstream from the first If the analyte is not detected in the second canister, then

it is safe to assume that the first canister’s capacity has not been exceeded The ume of gas that can be sampled before exceeding the sorbent’s capacity is called the

vol-breakthrough volume and is normally reported with units of m3/(gpack), where gpack

is the grams of sorbent

The short-term exposure of humans, animals, and plants to gaseous pollutants

is more severe than that for pollutants in other matrices Since the composition ofatmospheric gases can show a substantial variation over a time, the continuousmonitoring of atmospheric gases such as O3, CO, SO2, NH3, H2O2, and NO2by insitu sampling is important.10

Sample Preservation and Preparation After collecting the gross sample there isgenerally little need for sample preservation or preparation The chemical composi-tion of a gas sample is usually stable when it is collected using a solid sorbent, a fil-ter, or by cryogenic cooling When using a solid sorbent, gaseous compounds may

be removed before analysis by thermal desorption or by extracting with a suitablesolvent Alternatively, when the sorbent is selective for a single analyte, the increase

in the sorbent’s mass can be used to determine the analyte’s concentration in thesample

7C.3 Solids

Typical examples of solid samples include large particulates, such as those found inores; smaller particulates, such as soils and sediments; tablets, pellets, and capsulesused in dispensing pharmaceutical products and animal feeds; sheet materials, such

as polymers and rolled metals; and tissue samples from biological specimens

breakthrough volume

The volume of sample that can be passed

through a solid sorbent before the

analytes are no longer retained.

3

Sample Collection Solids are usually

heteroge-neous, and samples must be collected carefully if

they are to be representative of the target

popula-tion As noted earlier, solids come in a variety of

forms, each of which is sampled differently

Sediments from the bottom of streams, rivers,

lakes, estuaries, and oceans are collected with a

bot-tom grab sampler or with a corer Grab samplers are

equipped with a pair of “jaws” that close when they

contact the sediment, scooping up sediment in the

process (Figure 7.5) Their principal advantages are

ease of use and the ability to collect a large sample

Disadvantages include the tendency to lose

finer-grained sediment particles as water flows out of the

sampler and the loss of spatial information, both

lat-erally and with depth, due to mixing of the sample

An alternative method for sampling sediments

uses a cylindrical coring device (Figure 7.6) The

corer is dropped into the sediment, collecting a

col-umn of sediment and the water in contact with the

sediment With the possible exception of sediment

at the surface, which may experience mixing,

sam-ples collected with a corer maintain their vertical

profile As a result, changes in the sediment’s

com-position with depth are preserved The main

disad-vantage to a corer is that only a small surface area is

sampled For this reason sampling with a corer

usu-ally requires more samples

Soil samples collected at depths of up to 30 cm are easily collected with

scoops or shovels, although the sampling variance is generally high A better

method for obtaining soil samples near the surface is to use a soil punch This

thin-walled steel tube retains a core sample when it is pushed into the soil and

removed Soil samples collected at depths greater than 30 cm are obtained by

digging a trench and collecting lateral samples with a soil punch Alternatively,

Open

Closed

Sedimentar

y layersSea bottom

Figure 7.6

Schematic diagram of a piston corer in operation The weight of the corer is sufficient to cause its penetration into the sediment, while the upward motion of the piston allows water pressure to help force the sediment column into the barrel of the corer.

Figure 7.8

Schematic diagram of a sample thief.

Rotating the inner cylinder opens and closes

the openings along the outer cylinder’s

par-on opposite sides of the riffle Particulate material dumped into a riffle is dividedinto two parts By repeatedly passing half of the separated material back through theriffle, a sample of any desired size may be collected Smaller particulate materials,such as powders, are best collected with a sample thief, which allows material to becollected simultaneously from several locations (Figure 7.8) A typical sample thiefconsists of two tubes that are nestled together Each tube has an identical set of slotsaligned down their length Before the sample thief is inserted into the material beingsampled, the inner tube is rotated so that slots are closed When the sample thief is

in place, the inner tube is rotated to open the slots, allowing the powder to enter thesample thief through each slot The inner tube is then rotated to the closed positionand the sample thief withdrawn

When sampling a metal, it usually is necessary to obtain material from both thesurface and the interior When the metal is in the form of a sheet, random samplescan be collected with a metal punch Samples can be obtained from a metal wire byrandomly cutting off pieces of an appropriate length Larger pieces of metal, such asbars or bricks, are best sampled by sawing through the metal at randomly selectedpoints and collecting the “sawdust” or by drilling through the metal and collectingthe shavings A surface coating can be sampled in situ or by dissolving the coatingwith an appropriate solvent

Sampling of biological tissue is done by removing the entire organ, which isthen homogenized before smaller portions are taken for analysis Alternatively, sev-eral small portions of tissue may be combined to form a composite sample Thecomposite sample is then homogenized and analyzed

Sample Preservation Without preservation, many solid samples are subject tochanges in chemical composition due to the loss of volatile material, biodegrada-tion, and chemical reactivity (particularly redox reactions) Samples stored at re-duced temperatures are less prone to biodegradation and the loss of volatile mate-rial, but fracturing and phase separations may present problems The loss of volatilematerial is minimized by ensuring that the sample completely fills its containerwithout leaving a headspace where gases can collect Samples collected from mate-rials that have not been exposed to O2are particularly susceptible to oxidation reac-tions For example, the contact of air with anaerobic sediments must be prevented

Sample Preparation Unlike gases and liquids, which generally require little samplepreparation, solid samples usually need some processing before analysis There are tworeasons for this First, as discussed in Section 7B.3, sampling variance is a function ofthe number of particles sampled, not their combined mass For extremely heteroge-neous populations consisting of large particulates, the gross sample may be too large toanalyze For example, a boxcar containing a load of a Ni-bearing ore with an averageparticle size of 5 mm may require a sample weighing one ton to obtain a reasonablesampling variance Reducing the sample’s average particle size allows the same number

of particles to be sampled with a smaller, more manageable combined mass

Second, the majority of analytical techniques, particularly those used for aquantitative analysis, require that the analyte be in solution Solid samples, or atleast the analytes in a solid sample, must be brought into solution

Figure 7.7

Example of a four-unit riffle A sample

added through the top is divided into four

piles, two on each of the riffle’s sides.

Figure 7.9

Illustration showing the method of coning and quartering as a means of reducing a gross sample for subsampling (a) The gross sample is first piled into a cone and (b) flattened Looking down from above, (c) the cone is divided into four quarters, (d) which are then separated.

Reducing Particle Size A reduction in particle size is accomplished by a

combina-tion of crushing and grinding the gross sample The resulting particulates are then

thoroughly mixed and divided into samples of smaller mass containing the

appro-priate number of particles The process seldom occurs in a single step Instead,

sam-ples are cycled through the process several times until a laboratory sample of

de-sired mass is obtained

Crushing and grinding uses mechanical force to break larger particles into

smaller ones A variety of tools are used depending on the particle’s size and

hard-ness Large particles are crushed using jaw crushers capable of reducing particles to

diameters of a few millimeters Ball mills, disk mills, and mortars and pestles are

used to further reduce particle size

Significant changes in composition may occur during crushing and grinding

Decreasing particle size increases available surface area With more surface area

there is a greater risk of losing volatile components, a problem made worse by the

frictional heat accompanying the crushing and grinding An increase in surface area

also means that portions of the sample are freshly exposed to the atmosphere where

oxidation may alter the sample’s composition Other problems include

contamina-tion from the mechanical abrasion of the materials used to crush and grind the

sample, and differences in the ease with which particles are reduced in size Softer

particles are reduced in size more easily and may be lost as dust before the rest of

the sample has been processed This is a problem since the analyte’s distribution

may not be uniform between particles of different size

To ensure that all particles are reduced to a uniform size, the sample is

intermit-tently passed through a sieve Processing of those particles not passing through the

sieve continues until the entire sample is of uniform size The sample is then mixed

thoroughly to ensure homogeneity, and a secondary sample obtained with a riffle or

by coning and quartering The latter approach is outlined in Figure 7.9 The gross

sample is piled into a cone, flattened, divided into four quarters, and two diagonally

coning and quartering

A process for reducing the size of a gross sample.

With samples that are difficult to dissolve, the first approach is usually to try gesting the sample with an acid or base Table 7.2 lists the most commonly usedacids and bases and summarizes their use Digestion is commonly carried out in anopen container, such as a beaker, using a hot plate as a source of heat The chief ad-vantage of this approach is its low cost as it requires no special equipment Volatilereaction products, however, are lost, leading to a determinate error if analyte is in-cluded among the volatile substances.

di-Many digestions are now carried out in closed containers using microwave diation as a source of energy for heating the solution Vessels for microwave diges-tion are manufactured using Teflon (or some other fluoropolymer) or fused silica.Both materials are thermally stable, chemically resistant, transparent to microwaveradiation, and capable of withstanding elevated pressures A typical microwave di-gestion vessel is shown in Figure 7.10 and consists of the vessel body and cap, a tem-perature probe, and a pressure relief valve Vessels are placed in a microwave oven(typically 6–12 vessels can be accommodated), and microwave energy is controlled

ra-by monitoring the temperature or pressure within the vessels A microwave tion has several important advantages over an open container digestion, includinghigher temperatures (200–300 °C) and pressures (40–100 bar) As a result, diges-tions requiring several hours in an open container may be accomplished in less than

diges-Table 7.2 Acids and Bases Used for Sample Digestion

Solution

HCl (37%) • dissolves metals more easily reduced than H2(Eo < 0)

• dissolves insoluble carbonates, sulfides, phosphates, fluorides, sulfates, and many oxides

• dissolves most common metals except Al and Cr

• decomposes organics and biological samples (wet ashing)

• decomposes organics by oxidation and dehydration

HClO4 (70%) • hot, concentrated solutions are strong oxidizing agents

• dissolves many metals and alloys

• decomposes organics (reactions with organics are often

explosive, use only in specially equipped hoods with a blast shield and after prior decomposition with HNO3)

HCl:HNO3 (3:1 v/v) • also known as aqua regia

• dissolves Au and Pt

Pressure relief valve

probe

Cap

Vessel body

Table 7.3 Common Fluxes for Decomposing Inorganic Samples

Melting

30 min using microwave digestion In addition, the closed container prevents the

loss of volatile gases Disadvantages include the inability to add reagents during

di-gestion, limitations on the amount of sample that can be used (typically 1 g or less),

and safety concerns due to the use of high pressures and corrosive reagents

Appli-cations include environmental and biological samples

Inorganic samples that resist decomposition by digestion with acids or

bases often can be brought into solution by fusing with a large excess of an

al-kali metal salt, called a flux The sample and flux are mixed together in a

cru-cible and heated till the substances fuse together in a molten state The resulting

melt is allowed to cool slowly to room temperature Typically the melt dissolves

readily in distilled water or dilute acid Several common fluxes and their uses

are listed in Table 7.3 Fusion works when other methods of decomposition do

not because of the higher temperatures obtained and the high concentration of

the reactive flux in the molten liquid Disadvantages include a greater risk of

contamination from the large quantity of flux and the crucible and the loss of

volatile materials

Finally, organic materials may be decomposed by dry ashing In this method

the sample is placed in a suitable crucible and heated over a flame or in a furnace

Any carbon present in the sample is oxidized to CO2, and hydrogen, sulfur, and

ni-trogen are removed as H2O, SO2and N2 These gases can be trapped and weighed to

determine their content in the organic material Often the goal of dry ashing is the

removal of organic material, leaving behind an inorganic residue, or ash, that can be

further analyzed

7D Separating the Analyte from Interferents

When a method shows a high degree of selectivity for the analyte, the task of

per-forming a quantitative, qualitative, or characterization analysis is simplified For

ex-ample, a quantitative analysis for glucose in honey is easier to accomplish if the

method is selective for glucose, even in the presence of other reducing sugars, such

as fructose Unfortunately, analytical methods are rarely selective toward a single

species

In the absence of interferents, the relationship between the sample’s signal,

Ssamp, and the concentration of analyte, CA, is

*In equation 7.9, and the equations that follow, the concentration of analyte, CA , can be replaced by the moles of

analyte, nA , when considering a total analysis technique.

recovery

The fraction of analyte or interferent

remaining after a separation (R).

where kAis the analyte’s sensitivity.* In the presence of an interferent, equation 7.9becomes

where kIand CIare the interferent’s sensitivity and concentration, respectively Amethod’s selectivity is determined by the relative difference in its sensitivity toward

the analyte and interferent If kAis greater than kI, then the method is more

selec-tive for the analyte The method is more selecselec-tive for the interferent if kIis greater

than kA.Even if a method is more selective for an interferent, it can be used to deter-

mine an analyte’s concentration if the interferent’s contribution to Ssampis

insignifi-cant The selectivity coefficient, KA,I, was introduced in Chapter 3 as a means ofcharacterizing a method’s selectivity

con-KA,I×CI<< CA

When an interferent cannot be ignored, an accurate analysis must begin by ing the analyte and interferent

separat-7E General Theory of Separation Efficiency

The goal of an analytical separation is to remove either the analyte or the interferentfrom the sample matrix To achieve a separation there must be at least one signifi-cant difference between the chemical or physical properties of the analyte and inter-ferent Relying on chemical or physical properties, however, presents a fundamentalproblem—a separation also requires selectivity A separation that completely re-moves an interferent may result in the partial loss of analyte Altering the separation

to minimize the loss of analyte, however, may leave behind some of the interferent

A separation’s efficiency is influenced both by the failure to recover all the

ana-lyte and the failure to remove all the interferent We define the anaana-lyte’s recovery,

RA, as

where CAis the concentration of analyte remaining after the separation, and (CA)o

is the analyte’s initial concentration A recovery of 1.00 means that none of the

ana-lyte is lost during the separation The recovery of the interferent, RI, is defined inthe same manner

=

where CIis the concentration of interferent remaining after the separation, and

(CI)ois the interferent’s initial concentration The degree of separation is given by a

separation factor, SI,A, which is the change in the ratio of interferent to analyte

caused by the separation.11

EXAMPLE 7.10

An analysis to determine the concentration of Cu in an industrial plating bath

uses a procedure for which Zn is an interferent When a sample containing

128.6 ppm Cu is carried through a separation to remove Zn, the concentration

of Cu remaining is 127.2 ppm When a 134.9-ppm solution of Zn is carried

through the separation, a concentration of 4.3 ppm remains Calculate the

recoveries for Cu and Zn and the separation factor

SOLUTION

The recoveries for the analyte and interferent are

and

The separation factor is

In an ideal separation RA= 1, RI= 0, and SI,A= 0 In general, the separation factor

should be approximately 10–7for the quantitative analysis of a trace analyte in the

presence of a macro interferent, and 10–3when the analyte and interferent are

pres-ent in approximately equal amounts

Recoveries and separation factors are useful ways to evaluate the effectiveness

of a separation They do not, however, give a direct indication of the relative error

introduced by failing to remove all interferents or failing to recover all the analyte

The relative error introduced by the separation, E, is defined as

I, A I A

I o A o

I A

which simplifies to

7.16

A more useful equation for the relative error is obtained by solving equation 7.13

for CIand substituting back into equation 7.16

SCu= 1250×(ppm Cu)

SZn= 2310×(ppm Zn)(a) What error is expected if no attempt is made to remove Zn before analyzingfor Cu? (b) What is the error if the separation is carried out? (c) What is themaximum acceptable recovery for Zn if Cu is completely recovered and theerror due to the separation must be no greater than 0.10%?

SOLUTION

(a) If the analysis is carried out without a separation, then RCuand RZn areequal to 1.000, and equation 7.17 simplifies to

From equation 7.11 the selectivity coefficient is

Although we do not know the actual concentrations of Zn or Cu in the sample,

we do know that the concentration ratio (ppm Zn)o/(ppm Cu)ois 1/7 Thus

(b) To calculate the error, we substitute the recoveries calculated in Example7.10 into equation 7.17

C

C C